Some ideas and suggestions (warning: most of this is rambling. Some might be sensible, others might be ridiculous or just far-fetched. Also, no spell check, and you may suffer from a severe case of what is commonly known as "wall-of-text-disease" ):
-Combat: You said something about pets attacking after the player. But why not have a speed stat that determines who goes first? (kind of like pokemon I guess, haha) Could even have different stances that effect the stats, or different positions that things assume after taking certain actions (maybe something similar to the kerkerkruip dodge/parry/block mechanic, where your stats are effected based on what actions you've been taking lately). Maybe even skills that can only be used at specific points in the battle e.g: riposting attack.
-Chargen: I personally find the class gameplay thing a bit stereotypical, to be honest. I don't mean it as an insult to your games, or to any game at all, it's just that, in my personal opinion, it's been used so often that seeing a game that says "22 classes and 14 playable races!" in its descriptions just gets me listing all the stock D&D races in my head and, more often than not, moving on. So why not have skills determined by stats, instead of class. Players could be any mix of guns and magic, swords and bows, as long as they're willing to get the required stats high enough to get the set of skills they want. I mean what if I find a sword that can shoot lasers (somehow)? I'll obviously need a high swordwielding skill to wield it, but also some skills relating to guns and accuracy to fire the lasers. Or if I find a giant warhammer of destruction with runes of void generation. I'd need to know enough about runes to know how to activate them and enough about magic to properly use it, but also enough about warhammers, smashing and fighting techniques to swing the weapon around with some measure of skill. Etc etc.
-Pets: Players could find different pets in different places. You probably wouldn't find a cat in the ocean, for example. Pets could develop their own skills, and if they have the proper limbs, wear and wield things. If you're planning on mutations, pets could even show up with mutations (not always beneficial).
-Exploration: Something that has always gotten me about time travel games is how little the game lets you explore. You just enter the machine, push a button and it automatically knows where you want to go, or presents with a list of places that you have to go. I've always thought to myself: "Well why can't I go two years after this point, or to year x, or to whatever point in time I want?" I'd find it awesome if the machine would let you got to a whole bunch of different points in time, not all necessarily having to do with the storyline of the game. What if I want to explore the temperate region of asdfghjkl when asdfghjkl was known for being the hottest desert on the planet? Would be a great mechanic for finding new pets and monsters and equipment, too.
-Time: Different time periods could have different specialties, and players could be given different titles based on their skills depending on where they are -- a person who uses computers and high-tech machines might be known as a scientist in the future, but people in the far far past might refer to him as a sorcerer. "Any sufficiently advanced technology is indistinguishable from magic." Maybe even trigger different reactions from people -- this same scientist might be well excepted in the present, but maybe not so much in the superstition-plagued land of Azxcvbnmpot. Back to the "specialties" thing, there were probably empires that relied highly upon magic in the past, but maybe not so much in the future. There would be inquisition-like time periods where the player would find the best lore about elfslaying, because that's what the industry was thriving on at that time (elf-hide boots, muahahaha).
-Realtime impact: It would be nice to see the influence of changes you make in the past immediately (or not so immediately, but maybe in the way people react to you, what they do) in the future, and not just at the end of the game. Maybe a whole species of butterflies is wiped out, or evolves differently, because you killed one in the past, or a forest never covers the eastern region of Kerferferblumkptlglgland because you chopped down a tree in the prehistoric age. Maybe not so detailed, now that I think about it... coding all that would be a nightmare for decades, but similar mechanics on huge stuff. I'd understand not doing it for every tree and item, but if a player blows up half a mountain side or sinks the continent of R'lyeh...
-Knowledge: Since you're already implementing time travel as a core concept, it would be neat to see the descriptions of things change as you learn more by traveling through time. Like you might find a mountain at the beginning of the game called "Pflrogiurlkjvvoi mountain." The description might be "This is a very tall mountain, known for its frigid climate, savage wildlife, and the harsh, unforgiving stone of its slopes." Then you go back in time several thousand years and go to that same mountain. "You remember this to be near Pflrogiurlkjvvoi. It is just as harsh and cold as you remember it, although maybe a little shorter. A colony of dwarven miners lives here, they refer to the mountain as Mt. Ironcrag." Then you go back to the future (ha ha) and it says The same description pasted above, but then at the end: "In ancient times, this mountain used to be home to a civilization of dwarves, who called the mountain Mt. Ironcrag."
-Classes and skills: Depending on how you plan on going about it, there could be skills or classes that you can only unlock at certain time periods, or skills/spells/techniques you can only find in certain places, so players have to explore the entire world to make sure their class is as powerful as it can be. There might be a class that everyone thinks is horrible, they scream at you for ever creating it, and you just grin evilly and rub your hands together, until someone says "woah! what's this!" and finds a skill in some hidden cave for that class. Suddenly everyone wants to play that class again and explore the game more, hunting for hidden skills.
OK... This is probably waaaay too long already, so I'll stop rambling. Thanks for reading, have a nice day, scream at me for making so many impossible suggestions, etc.
Go, balloons. I don't see anything happening. Go, balloons. Go, balloons. Go, balloons. Stand by, confetti. Keep coming, balloons. More balloons. Bring them. Balloons, balloons, balloons! More balloons. Tons of them. Bring them down. Let them all come. No confetti. No confetti yet. No confetti. All right. Go, balloons.
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